Five months after going public in one of the most controversial IPOs in U.S. history, social networking titan Facebook finally delivered some good news on Tuesday, reporting earnings results that impressed Wall Street. With its share price down nearly 50% since the IPO, Facebook has become the quintessential “show-me stock,” with spooked investors sitting on the sidelines waiting for the company to make a compelling case. Facebook is not out of the woods, but solid third-quarter revenue growth and surprisingly positive progress in the mobile space caused the company’s shares to soar nearly 13% in after-hours trading.

That big “whooshing” sound? It’s Facebook executives and shareholders exhaling. After a very rocky first half of the year, Facebook appears to be getting its groove back.

Overall revenue increased 32% in the quarter to $1.26 billion, beating analyst expectations. But what really jazzed Wall Street was Facebook’s disclosure that mobile revenue accounted for 14% of the company’s ad sales, far exceeding what analysts had been predicting. “Advertising revenue from mobile was the number that really stood out,” Sterne, Agee & Leach analyst Arvind Bhatia toldReuters. “They are saying mobile ad revenue was 14% of total ad revenue. That would be about $140 million and I was expecting $40 or $50 million from this.”

Facebook’s mobile progress is absolutely crucial, because that area had been viewed as the company’s Achilles heel. In the weeks leading up to Facebook’s IPO, the company warned that it might be more difficult to make money off mobile users, as compared to those who access the service through its traditional desktop platform. On a conference call with Wall Street analysts, CEO Mark Zuckerberg said he wanted to “dispel the myth” that Facebook can’t make money in the mobile space. “This may have seemed true earlier this year because we hadn’t started trying yet,” Zuckerberg said.

Mobile revenue is rapidly becoming the holy grail of consumer Internet companies. That’s because the industry is undergoing a profound and massive structural shift thanks to the explosive growth of sophisticated smartphones like Apple‘s iPhone and Google’s Android devices. Simply put, users are increasingly accessing the Internet on mobile devices, and there is a general consensus in the industry that the locus of computing is shifting away from the desktop and toward smartphones and tablets.

Facebook now has 1 billion users worldwide who access the company’s platform every month, and nearly 600 million who visit the site every day. Although the company posted a net loss of $59 million, analysts didn’t seem to mind, given strong revenue growth, progress in mobile, and the company’s vast and growing audience.

“Our daily reach is more than 3 times larger than the total viewing audience of the Super Bowl, and it happens every day,” Facebook COO Sheryl Sandberg told analysts. Needless to say, the company’s advertising opportunities are immense, but Zuckerberg has been wary about cluttering up the website’s user interface with intrusive ads. Facebook now is slowly moving to place more ads in the News Feed, the central stream of updates on a user’s profile page. “We’ve been very focused on getting more ads into news feeds — it’s one of our primary priorities,” Sandberg toldThe Wall Street Journal.

Zuckerberg touted the company’s mobile progress. “As proud as I am that a billion people use Facebook each month, I’m also really happy that over 600 million people now share and connect on Facebook every month using mobile devices,” he said. “People who use our mobile products are more engaged, and we believe we can increase engagement even further as we continue to introduce new products and improve our platform.”

We are hurtling toward a day when billions of people will be carrying around Web-enabled mobile computers that are more powerful than the most robust desktop computers of just a few years ago. Facebook is well positioned in this area, because its mobile application is the most popular app on the major mobile platforms, including Apple’s iOS and Google’s Android. Facebook executives said the company has “completely rebuilt” its iOS application for “faster and more reliable performance,” and CFO David Ebersman told analysts that the company “opened up a lot of [advertising] inventory on mobile in this quarter, so we had advertisers shift to mobile feeds.”

Despite Tuesday’s good news, Facebook is still dealing with the lingering fallout from its controversial IPO. The offering was supposed to be a triumphant moment for Silicon Valley and Wall Street. Instead, between trading glitches on the NASDAQ exchange, accusations of “selective disclosure” against Facebook’s bankers, and an offering level that was clearly mis-priced, the episode turned into a major debacle.

After going public at $38 per share, Facebook’s stock price dropped like a rock to about $20, where it’s been hovering ever since. The company and its bankers face multiple lawsuits from disgruntled investors, but in another spot of good news, the Securities and Exchange Commission has thus far found no evidence of wrongdoing by Facebook, although the investigation is continuing.

Facebook still has a ton of work to do — especially if it intends to push its stock anywhere near the IPO price — but Tuesday’s results indicate the company is making progress with its strategy to generate revenue from mobile advertising, which is the next great tech battlefield in the war for Internet supremacy.